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The Chebyshev polynomials of the first kind $T_n(x)$ are known to form a complete orthogonal basis for functions on $[-1,1]$. I was looking for a proof of the completeness part, without any luck, when I finally bumped into this formula on Wolfram's function site: $$ \sum_{n=0}^\infty T_n(x)T_n(y)= \frac{\pi}{2}\sqrt[4]{1-x^2}\sqrt[4]{1-y^2}\delta(x-y),\quad x,y\in(-1,1). $$ Does anybody know how to prove this? Or can you please suggest a book where I can find it?

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  • $\begingroup$ To prove completeness it suffices to prove that the Chebyshev polynomials are dense (with respect to the appropriate $L^2$ metric, the one wrt to which the Chebyshev polynomials are orthogonal). The subspace they span is the subspace of all polynomials, so it suffices to prove that polynomials are dense. And for this you can appeal to Stone-Weierstrass. $\endgroup$ – Qiaochu Yuan Dec 29 '17 at 23:30
  • $\begingroup$ How would the above be used to show completeness? $\endgroup$ – copper.hat Dec 29 '17 at 23:31
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    $\begingroup$ @copper.hat The delta function on the right hand side is essentially the identity operator on the function space, so this formula gives a representation of the identity map in terms on Chebyshev polynomials. These formulas are commonly referred to as "completeness relations". You can find many similar expressions for other orthogonal polynomials, e.g. Laguerre here $\endgroup$ – Andras Vanyolos Dec 29 '17 at 23:41

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